ntity; for a
body starting with twice the velocity of another, will rise to four
times the height. In like manner, a three-fold velocity will give a
nine-fold elevation, a four-fold velocity will give a sixteen-fold
elevation, and so on. The height attained, then, is not proportional
to the initial velocity, but to the square of the velocity. As
before, the work is also proportional to the weight elevated. Hence
the work which any moving mass whatever is competent to perform, in
virtue of the motion which it at any moment possesses, is jointly
proportional to its weight and the square of its velocity. Here,
then, we have a second measure of work-, in which we simply translate
the idea of height into its equivalent idea of motion.

In mechanics, the product of the mass of a moving body into the square
of its velocity, expresses what is called the _vis viva_, or living
force. It is also sometimes called the 'mechanical effect.' If, for
example, a cannon pointed to the zenith urge a ball upwards with twice
the velocity imparted to a second ball, the former will rise to four
times the height attained by the latter. If directed against a
target, it will also do four times the execution. Hence the
importance of imparting a high velocity to projectiles in war. Having
thus cleared our way to a perfectly definite conception of the _vis
viva_ of moving masses, we are prepared for the announcement that the
heat generated by the shock of a falling body against the earth is
proportional to the _vis viva_ annihilated. The heat is proportional to
the square of the velocity. In the case, therefore, of two
cannon-balls of equal weight, if one strike a target with twice the
velocity of the other, it will generate four times the heat, if with
three times the velocity, it will generate nine times the heat, and so
on.

Mr. Joule has shown that a pound weight falling from a height of 772
feet, or 772 pounds falling through one foot, will generate by its
collision with the earth an amount of heat sufficient to raise a pound
of water one degree Fahrenheit in temperature. 772 "foot-pounds"
constitute the mechanical equivalent of heat. Now, a body falling
from a height of 772 feet, has, upon striking the earth, a velocity of
223 feet a second; and if this velocity were imparted to the body, by
any other means, the quantity of heat generated by the stoppage of its
motion would be that stated above. Six times that velocity, or 1,33